Ample Divisors, Automorphic Forms and Shafarevich's Conjecture

TL;DR

This paper develops a general method to prove finiteness results for families of polarized algebraic varieties over Riemann surfaces, using properties of their moduli spaces and discriminant divisors, with applications to K3 and Enriques surfaces.

Contribution

It introduces a new approach linking the ampleness of discriminant divisors in moduli spaces to the finiteness of algebraic variety families, extending Shafarevich's conjecture.

Findings

Finiteness of families of certain polarized algebraic varieties established.

Method applied successfully to K3 and Enriques surfaces.

Ampleness of discriminant divisor implies Shafarevich-type finiteness.

Abstract

In this article we give a general approach to the following analogue of Shafarevich's conjecture for some polarized algebraic varieties; suppose that we fix a type of an algebraic variety and look at families of such type of varieties over a fixed Riemann surface with fixed points over which we have singular varieties, then one can ask if the set of such families, up to isomorphism, is finite. In this paper we give a general approach to such types of problems. The main observation is the following; suppose that the moduli space of a fixed type of algebraic polarized variety exists and suppose that in some projective smooth compactification of the coarse moduli the discriminant divisor supports an ample one, then it is not difficult to see that this fact implies the analogue of Shafarevich's conjecture. In this article we apply this method to certain polarized algebraic K3 surfaces and also to Enriques surfaces.